We now turn to interacting many-body cases.
An important feature is that a typical short-range interaction in the \(x\) -\(y\)  plane translates into long-ranged and unconventional interaction in the \(p_y\)  direction.
In particular, the formation of vortices, which is a characteristic feature of weakly interacting bosons in a magnetic field [1]}, [2]}, [3]}, [4]}, [5]}, cannot be observed in the \(x\) -\(p_y\)  plane only with a contact interaction in the \(x\) -\(y\)  plane.
Nonetheless, we find that, by adopting a long-range interaction in the \(y\)  direction, vortices can still form in our setup as we discuss below.
We now focus on bosons with the mean-field interaction and ask if vortices can form in \(x\) -\(p_y\)  plane.
We use description in terms of the Gross-Pitaevskii equation with the condensate wave function \(\Psi _n (x,t)\) :
\(&i\partial _t \Psi _n (x,t) = H_\mathrm {yhop} + H_{\mathrm {int}} + \\&\left\lbrace  -\frac{\partial _x^2}{2m} + \frac{1}{2}m\omega ^2 (x - n x_0)^2+ \frac{1}{2}m\tilde{\omega }^2 x^2\right\rbrace  \Psi _n (x,t), \\&H_\mathrm {yhop} = J_y (\Psi _{n-1}(x,t) + \Psi _{n+1}(x,t)), \\&H_\mathrm {int} = g_s |\Psi _n (x,t)|^2 \Psi _n (x,t) \\ &\hspace{28.45274pt} + g_1 (|\Psi _{n-1} (x,t)|^2 + |\Psi _{n+1} (x,t)|^2) \Psi _n (x,t).\) 
